Why do Americans want to know the month first and the day second?
I'm sorry but it doesn't make sense TO ME. Based on what I was taught, regardless of the month, I think what matters first is to know what day of the month you are in, if at the beginning, in the middle or at the end of said month. After you know that, you can find out the month to know where you are in the year.
What is the benefit of doing it the other way around?
In UK English, it's considered proper to write "the 6th of March" as "6 March" and sometimes read as "6th March" which can be jarring to Americans as their shorthand is "March 6th" and when "6(th) March" is encountered in written form, it's expanded to the full "6th of March" when spoken
That doesn't mean this won't be yet another feature American English absorbs from UK English but right now flipping them in speech requires a few extra syllables and people are lazy